The technology of Digital Subscriber Line (DSL) is a high speed transmission technology in which data is transmitted through a twist pair which is designed for voice transmission, i.e. Unshielded Twist Pair (UDP). The DSL includes the Asymmetrical Digital Subscriber Line (ADSL), the Very-high-bit-rate Digital Subscriber Line (VDSL), the Digital Subscriber Line (IDSL) based upon the Integrated Services Digital Network (ISDN), and the Single-pair High-bit-rate Digital Subscriber Line (SHDSL), etc.
With the increase of frequency bands used for the various digital subscriber line technologies (xDSL), the issue of crosstalk, especially the issue of crosstalk in high frequency bands becomes apparent more and more. As illustrated in FIG. 1, since the frequency division multiplexing is adopted in downlink and uplink channels of xDSL, the near-end crosstalk (NEXT) will not cause much damage to the performance of a system, but the far-end crosstalk (FEXT) will influence the transmission performance of the lines severely. When a plurality of users out of a bundle of cables require fulfillment of an xDSL service, some lines will be of low rate and unstable performance, and even not be fulfilled with required services due to the far-end crosstalk, which ultimately leads to a low line activation ratio of a DSLAM (Digital Subscriber Line Access Multiplexer).
In a communication model in which Discrete Multi-Tone Modulation (DMT) is adopted and K users and N Tones are included, the signal transmission over each Tone can be expressed independently as follows:yn=Hnxn+σn 
Normally, the receiving end of each xDSL modem takes interference from other modems as noise, and the data rate achieved by the kth user over the nth Tone can be calculated with the Shannon channel capacity formula as follows:
      b    n    k    =            log      2        (          1      +                                                                                h                n                                  k                  ,                  k                                                                    2                    ⁢                      s            n            k                                                              ∑                              j                ≠                k                                      ⁢                                                                                                  h                    n                                          k                      ,                      j                                                                                        2                            ⁢                              s                n                j                                              +                      σ            k                                )  
As can be seen from the above formula, the crosstalk will bring a serious influence upon the transmission capacity of a line, that is, reduce the line rate.
The Dynamic Spectrum Management (DSM) can be used to reduce the influence of the crosstalk. Specifically, the DSM is intended to reduce the crosstalk by automatically adjusting transmission power over each modem in a network. Particularly in the case of a CO/RT hybrid application, the influence of the crosstalk of a short line upon a long line may be more serious. The object of the DSM is to adjust the transmission power for each modem to achieve a balance between maximizing its rate and reducing influence of its crosstalk upon other modems.
The goal of the DSM is that in the case of total transmission power of each user not exceeding a threshold, value of transmission power of each user in each sub-carrier is adjusted so as to maximize a weighted rate sum of all the users. Consequently, the DSM solution can be expressed mathematically as follows:
      Maximize    ⁢                  ⁢                  ∑                  k          =          1                K            ⁢                        ω          k                ⁢                              ∑                          n              =              1                        N                    ⁢                      b            n            k                                                  Subject        ⁢                                  ⁢        to        ⁢                                  ⁢                              ∑                          n              =              1                        N                    ⁢                      S            n            k                              ≤              P        k              ,          ∀      k                  0      ≤                        S          n          k                ⁢                                  ⁢                  ∀          k                      ,    n  
Snk denotes the power allocated to the kth user in the nth sub-carrier;
Gnkk denotes a transmission coefficient of the kth subscriber line in the nth sub-carrier;
Gnkj(j≠k) denotes a crosstalk coefficient of the jth user in the nth sub-carrier with respect to the kth user;
Pk denotes a total power threshold of the kth user;
ωk denotes a rate weight coefficient of the kth user;
σ2 denotes noise power;
N denotes the total number of sub-carriers;
K denotes the total number of users.
This solution is a nonlinear constraint optimization solution, in which both the objective function and the constraint condition are non-convex functions of independent variables. Therefore, there is no efficient and complete solution algorithm. Among existing algorithms, OSB (Optimal Spectrum Balancing) and ISB (Iterative Spectrum Balancing) are the most popular ones. These two algorithms each change the original nonlinear constraint optimization problem to a nonlinear unconstraint optimization problem through an introduction of a Lagrange Multiplier. Thus, the above formula is converted into the follows:
      max    ⁢                  ∑                  k          =          1                K            ⁢                          ⁢                        ω          k                ⁢                              ∑                          n              =              1                        N                    ⁢                                          ⁢                      b            n            k                                +            ∑              k        =        1            K        ⁢                  ⁢                  λ        k            ⁡              (                              P            k                    -                                    ∑                              n                =                1                            N                        ⁢                                                  ⁢                          S              n              k                                      )            
λk is a Lagrange Multiplier, and the term
      ∑          k      =      1        K    ⁢          ⁢            λ      k        ⁡          (                        P          k                -                              ∑                          n              =              1                        N                    ⁢                                          ⁢                      S            n            k                              )      indicates a total power constraint
      (                                        ∑                          n              =              1                        N                    ⁢                                          ⁢                      S            n            k                          ≤                  P          k                    ,              ∀        k              )    .
Since Pk is a constant, the above formula is equivalent to the follows:
      max    ⁢                  ∑                  k          =          1                K            ⁢                          ⁢                        ω          k                ⁢                              ∑                          n              =              1                        N                    ⁢                                          ⁢                      b            n            k                                -            ∑              k        =        1            K        ⁢                  ⁢                  λ        k            ⁢                        ∑                      n            =            1                    N                ⁢                                  ⁢                  S          n          k                    
Thus, it will be sufficient to solve the above formula.
A method for implementing the DSM in the prior art is an Iterative Water-Filling (IWF) algorithm.
This is a greedy algorithm, which only takes into account the influence of snk upon the rate over the kth subscribe line, and if a current subscriber line is not of the kth user, it is assumed that wj=0,λj=0, ∀j≠k. In this way, Jn can be actually simplified as Jn≅Jnkwkbnk−λksnk, because at this time, wj=0,λj=0, ∀j≠k and snj(j≠k) is fixed. For a solution of snk=arg max Jnk, because there is a unique extremum point for Jnk=Wkbnk−λksnk, a solution expression can be derived for the extremum point. The entire algorithm can be iterated continuously in order to solve max Jnk for users over respective Tones until power allocation for all the users does not vary.
The IWF algorithm is less complicated, and can be applicable with relatively large N and K. Moreover, the algorithm is autonomic completely, i.e., users only need to optimize their rates and satisfy their power constraints respectively, and no exchange of data and information between different users or no central manager is needed. Therefore the algorithm can be readily implemented in a real system. However, the IWF is a greedy algorithm, and depends on an initial solution, so it can not ensure an optimal solution or an approximately optimal solution.